My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
April 30th, 2017, 02:39 AM   #1
Senior Member
 
Joined: Nov 2015
From: hyderabad

Posts: 154
Thanks: 1

Summation Problem

Hello Guys!
Can someone help me with this!

How many pair of positive integers of $(m,n)$ are there satisfying
$\sum_{i=1}^{n} i! = m!$

I guess it has only one pair of solution : $(m,n) =(1,1)$
Then starting and ending value of Summation should be equal as I start at $i=1$ and we need $n$ to be 1.

Will it make sense or Is it wrong to have both values same ?

Please let me know
Thank you
Lalitha183 is offline  
 
April 30th, 2017, 04:00 AM   #2
Math Team
 
agentredlum's Avatar
 
Joined: Jul 2011
From: North America, 42nd parallel

Posts: 3,277
Thanks: 204

For $ \ \ \ \ n > 1 \ \ \ \ $ , $ \ \ \ \ n < m \ \ \ \ $ otherwise equality has no chance. Is this obvious to you?

Suppose $ \ \ \ \ n = m - 1$

We need to determine if $ \ \ \ \ \sum_{i=1}^{m-1}i! \ \ \ \ $ can catch up to $ \ \ \ \ m!$

Let's see how the desired equality would look

$ 1! + 2! + 3! + ... + (m-1)! = 1 \times 2 \times 3 \times ... \times (m-1) \times m $

$ 1! + 2! + 3! + ... + (m-1)! = (m-1)! \times m $

$ 1! + 2! + ... + (m-1)! = (m-1)! + (m-1)! + ... + (m-1)! $

The left hand side has $ \ \ \ \ m-1 \ \ \ \ $ sums , the right hand side has $ \ \ \ \ m \ \ \ \ $ sums. Moreover , every sum on the left except the last sum is smaller than any sum on the right.

So no chance to catch up


Last edited by agentredlum; April 30th, 2017 at 04:06 AM.
agentredlum is offline  
April 30th, 2017, 04:29 AM   #3
Math Team
 
agentredlum's Avatar
 
Joined: Jul 2011
From: North America, 42nd parallel

Posts: 3,277
Thanks: 204

You can have both values the same

$ \sum_{i = 1}^{1} f(i) = f(1) $

agentredlum is offline  
April 30th, 2017, 04:35 AM   #4
Senior Member
 
Joined: Nov 2015
From: hyderabad

Posts: 154
Thanks: 1

Quote:
Originally Posted by agentredlum View Post
You can have both values the same

$ \sum_{i = 1}^{1} f(i) = f(1) $

Thank you
Lalitha183 is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
problem, summation



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Struggling with what I think is a summation problem. phileagleson Calculus 5 November 5th, 2016 11:31 AM
Basic summation problem squishyturtle Pre-Calculus 2 January 19th, 2015 08:16 PM
Annoying summation problem johnr Number Theory 3 May 7th, 2013 04:58 PM
summation problem colorless Algebra 7 October 6th, 2011 04:47 AM
summation problem mikeportnoy Algebra 1 February 11th, 2009 04:46 PM





Copyright © 2017 My Math Forum. All rights reserved.